principal bundle

Let $E$ be a topological space on which a topological group $G$ acts continuously and freely. The map $\pi :E\to E/G=B$ is called a principal bundle (or principal $G$-bundle) if the projectionmap $\pi :E\to B$ is a locally trivial bundle.

Any principal bundle with a section $\sigma :B\to E$ is trivial, since the map $\varphi :B\times G\to E$ given by $\varphi (b,g)=g\cdot \sigma (b)$ is an isomorphism. In particular, any $G$-bundle which is topologically trivial is also isomorphic to $B\times G$ as a $G$-space. Thus any local trivialization of $\pi :E\to B$ as a topological bundle is an equivariant trivialization.